L(s) = 1 | − 2-s − 3.14·3-s + 4-s − 2.17·5-s + 3.14·6-s + 4.77·7-s − 8-s + 6.90·9-s + 2.17·10-s + 0.464·11-s − 3.14·12-s − 3.54·13-s − 4.77·14-s + 6.84·15-s + 16-s + 2.17·17-s − 6.90·18-s − 7.81·19-s − 2.17·20-s − 15.0·21-s − 0.464·22-s − 2.67·23-s + 3.14·24-s − 0.276·25-s + 3.54·26-s − 12.2·27-s + 4.77·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.81·3-s + 0.5·4-s − 0.971·5-s + 1.28·6-s + 1.80·7-s − 0.353·8-s + 2.30·9-s + 0.687·10-s + 0.140·11-s − 0.908·12-s − 0.981·13-s − 1.27·14-s + 1.76·15-s + 0.250·16-s + 0.526·17-s − 1.62·18-s − 1.79·19-s − 0.485·20-s − 3.27·21-s − 0.0990·22-s − 0.558·23-s + 0.642·24-s − 0.0552·25-s + 0.694·26-s − 2.36·27-s + 0.901·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3501483418\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3501483418\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 4001 | \( 1+O(T) \) |
good | 3 | \( 1 + 3.14T + 3T^{2} \) |
| 5 | \( 1 + 2.17T + 5T^{2} \) |
| 7 | \( 1 - 4.77T + 7T^{2} \) |
| 11 | \( 1 - 0.464T + 11T^{2} \) |
| 13 | \( 1 + 3.54T + 13T^{2} \) |
| 17 | \( 1 - 2.17T + 17T^{2} \) |
| 19 | \( 1 + 7.81T + 19T^{2} \) |
| 23 | \( 1 + 2.67T + 23T^{2} \) |
| 29 | \( 1 + 4.66T + 29T^{2} \) |
| 31 | \( 1 + 2.75T + 31T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 41 | \( 1 - 0.752T + 41T^{2} \) |
| 43 | \( 1 - 0.446T + 43T^{2} \) |
| 47 | \( 1 - 6.33T + 47T^{2} \) |
| 53 | \( 1 + 5.24T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 6.29T + 61T^{2} \) |
| 67 | \( 1 + 7.37T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 2.14T + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 5.61T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69903475702647126925086440726, −7.30641864362545731124635931545, −6.53183327193618826168196003293, −5.68090824109715383381627270422, −5.14550039251633807784355677459, −4.38649255901985820421067083403, −3.94916395123625344301795292580, −2.18087023410271143343834259421, −1.48749134070606017195144576099, −0.37533032058123430276480051273,
0.37533032058123430276480051273, 1.48749134070606017195144576099, 2.18087023410271143343834259421, 3.94916395123625344301795292580, 4.38649255901985820421067083403, 5.14550039251633807784355677459, 5.68090824109715383381627270422, 6.53183327193618826168196003293, 7.30641864362545731124635931545, 7.69903475702647126925086440726